Integrand size = 168, antiderivative size = 30 \[ \int \frac {5-25 x-108 x^2-180 x^3+36 x^2 \log (x)+\left (15-11 x-60 x^2+(-5+12 x) \log (x)\right ) \log (3+5 x-\log (x))+(-3-5 x+\log (x)) \log ^2(3+5 x-\log (x))}{360 x^2+168 x^3-720 x^4+\left (-120 x^2+144 x^3\right ) \log (x)+\left (60 x-44 x^2-240 x^3+\left (-20 x+48 x^2\right ) \log (x)\right ) \log (3+5 x-\log (x))+\left (-12 x-20 x^2+4 x \log (x)\right ) \log ^2(3+5 x-\log (x))} \, dx=\frac {1}{4} \log \left (x-\frac {x}{x+\frac {1}{5} (x+\log (3+5 x-\log (x)))}\right ) \]
Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {5-25 x-108 x^2-180 x^3+36 x^2 \log (x)+\left (15-11 x-60 x^2+(-5+12 x) \log (x)\right ) \log (3+5 x-\log (x))+(-3-5 x+\log (x)) \log ^2(3+5 x-\log (x))}{360 x^2+168 x^3-720 x^4+\left (-120 x^2+144 x^3\right ) \log (x)+\left (60 x-44 x^2-240 x^3+\left (-20 x+48 x^2\right ) \log (x)\right ) \log (3+5 x-\log (x))+\left (-12 x-20 x^2+4 x \log (x)\right ) \log ^2(3+5 x-\log (x))} \, dx=\frac {1}{4} (\log (x)+\log (5-6 x-\log (3+5 x-\log (x)))-\log (6 x+\log (3+5 x-\log (x)))) \]
Integrate[(5 - 25*x - 108*x^2 - 180*x^3 + 36*x^2*Log[x] + (15 - 11*x - 60* x^2 + (-5 + 12*x)*Log[x])*Log[3 + 5*x - Log[x]] + (-3 - 5*x + Log[x])*Log[ 3 + 5*x - Log[x]]^2)/(360*x^2 + 168*x^3 - 720*x^4 + (-120*x^2 + 144*x^3)*L og[x] + (60*x - 44*x^2 - 240*x^3 + (-20*x + 48*x^2)*Log[x])*Log[3 + 5*x - Log[x]] + (-12*x - 20*x^2 + 4*x*Log[x])*Log[3 + 5*x - Log[x]]^2),x]
Time = 1.47 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {7292, 27, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {-180 x^3-108 x^2+36 x^2 \log (x)+\left (-60 x^2-11 x+(12 x-5) \log (x)+15\right ) \log (5 x-\log (x)+3)-25 x+(-5 x+\log (x)-3) \log ^2(5 x-\log (x)+3)+5}{-720 x^4+168 x^3+360 x^2+\left (-20 x^2-12 x+4 x \log (x)\right ) \log ^2(5 x-\log (x)+3)+\left (144 x^3-120 x^2\right ) \log (x)+\left (-240 x^3-44 x^2+\left (48 x^2-20 x\right ) \log (x)+60 x\right ) \log (5 x-\log (x)+3)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-180 x^3-108 x^2+36 x^2 \log (x)+\left (-60 x^2-11 x+(12 x-5) \log (x)+15\right ) \log (5 x-\log (x)+3)-25 x+(-5 x+\log (x)-3) \log ^2(5 x-\log (x)+3)+5}{4 x (5 x-\log (x)+3) \left (-36 x^2+30 x-\log ^2(5 x-\log (x)+3)-12 x \log (5 x-\log (x)+3)+5 \log (5 x-\log (x)+3)\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int \frac {-180 x^3+36 \log (x) x^2-108 x^2-25 x-(5 x-\log (x)+3) \log ^2(5 x-\log (x)+3)+\left (-60 x^2-11 x-(5-12 x) \log (x)+15\right ) \log (5 x-\log (x)+3)+5}{x (5 x-\log (x)+3) \left (-36 x^2-12 \log (5 x-\log (x)+3) x+30 x-\log ^2(5 x-\log (x)+3)+5 \log (5 x-\log (x)+3)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{4} \int \left (\frac {30 x^2-6 \log (x) x+23 x-1}{x (5 x-\log (x)+3) (6 x+\log (5 x-\log (x)+3)-5)}+\frac {1}{x}+\frac {-30 x^2+6 \log (x) x-23 x+1}{x (5 x-\log (x)+3) (6 x+\log (5 x-\log (x)+3))}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} (\log (x)+\log (-6 x-\log (5 x-\log (x)+3)+5)-\log (6 x+\log (5 x-\log (x)+3)))\) |
Int[(5 - 25*x - 108*x^2 - 180*x^3 + 36*x^2*Log[x] + (15 - 11*x - 60*x^2 + (-5 + 12*x)*Log[x])*Log[3 + 5*x - Log[x]] + (-3 - 5*x + Log[x])*Log[3 + 5* x - Log[x]]^2)/(360*x^2 + 168*x^3 - 720*x^4 + (-120*x^2 + 144*x^3)*Log[x] + (60*x - 44*x^2 - 240*x^3 + (-20*x + 48*x^2)*Log[x])*Log[3 + 5*x - Log[x] ] + (-12*x - 20*x^2 + 4*x*Log[x])*Log[3 + 5*x - Log[x]]^2),x]
3.18.38.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.48 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37
| method | result | size |
| default | \(\frac {\ln \left (x \right )}{4}+\frac {\ln \left (-5+6 x +\ln \left (-\ln \left (x \right )+5 x +3\right )\right )}{4}-\frac {\ln \left (6 x +\ln \left (-\ln \left (x \right )+5 x +3\right )\right )}{4}\) | \(41\) |
| risch | \(\frac {\ln \left (x \right )}{4}+\frac {\ln \left (-5+6 x +\ln \left (-\ln \left (x \right )+5 x +3\right )\right )}{4}-\frac {\ln \left (6 x +\ln \left (-\ln \left (x \right )+5 x +3\right )\right )}{4}\) | \(41\) |
| parallelrisch | \(\frac {3}{2}-\frac {\ln \left (x +\frac {\ln \left (-\ln \left (x \right )+5 x +3\right )}{6}\right )}{4}+\frac {\ln \left (x +\frac {\ln \left (-\ln \left (x \right )+5 x +3\right )}{6}-\frac {5}{6}\right )}{4}+\frac {\ln \left (x \right )}{4}\) | \(42\) |
int(((ln(x)-5*x-3)*ln(-ln(x)+5*x+3)^2+((12*x-5)*ln(x)-60*x^2-11*x+15)*ln(- ln(x)+5*x+3)+36*x^2*ln(x)-180*x^3-108*x^2-25*x+5)/((4*x*ln(x)-20*x^2-12*x) *ln(-ln(x)+5*x+3)^2+((48*x^2-20*x)*ln(x)-240*x^3-44*x^2+60*x)*ln(-ln(x)+5* x+3)+(144*x^3-120*x^2)*ln(x)-720*x^4+168*x^3+360*x^2),x,method=_RETURNVERB OSE)
Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int \frac {5-25 x-108 x^2-180 x^3+36 x^2 \log (x)+\left (15-11 x-60 x^2+(-5+12 x) \log (x)\right ) \log (3+5 x-\log (x))+(-3-5 x+\log (x)) \log ^2(3+5 x-\log (x))}{360 x^2+168 x^3-720 x^4+\left (-120 x^2+144 x^3\right ) \log (x)+\left (60 x-44 x^2-240 x^3+\left (-20 x+48 x^2\right ) \log (x)\right ) \log (3+5 x-\log (x))+\left (-12 x-20 x^2+4 x \log (x)\right ) \log ^2(3+5 x-\log (x))} \, dx=-\frac {1}{4} \, \log \left (6 \, x + \log \left (5 \, x - \log \left (x\right ) + 3\right )\right ) + \frac {1}{4} \, \log \left (6 \, x + \log \left (5 \, x - \log \left (x\right ) + 3\right ) - 5\right ) + \frac {1}{4} \, \log \left (x\right ) \]
integrate(((log(x)-5*x-3)*log(-log(x)+5*x+3)^2+((12*x-5)*log(x)-60*x^2-11* x+15)*log(-log(x)+5*x+3)+36*x^2*log(x)-180*x^3-108*x^2-25*x+5)/((4*x*log(x )-20*x^2-12*x)*log(-log(x)+5*x+3)^2+((48*x^2-20*x)*log(x)-240*x^3-44*x^2+6 0*x)*log(-log(x)+5*x+3)+(144*x^3-120*x^2)*log(x)-720*x^4+168*x^3+360*x^2), x, algorithm=\
-1/4*log(6*x + log(5*x - log(x) + 3)) + 1/4*log(6*x + log(5*x - log(x) + 3 ) - 5) + 1/4*log(x)
Exception generated. \[ \int \frac {5-25 x-108 x^2-180 x^3+36 x^2 \log (x)+\left (15-11 x-60 x^2+(-5+12 x) \log (x)\right ) \log (3+5 x-\log (x))+(-3-5 x+\log (x)) \log ^2(3+5 x-\log (x))}{360 x^2+168 x^3-720 x^4+\left (-120 x^2+144 x^3\right ) \log (x)+\left (60 x-44 x^2-240 x^3+\left (-20 x+48 x^2\right ) \log (x)\right ) \log (3+5 x-\log (x))+\left (-12 x-20 x^2+4 x \log (x)\right ) \log ^2(3+5 x-\log (x))} \, dx=\text {Exception raised: PolynomialError} \]
integrate(((ln(x)-5*x-3)*ln(-ln(x)+5*x+3)**2+((12*x-5)*ln(x)-60*x**2-11*x+ 15)*ln(-ln(x)+5*x+3)+36*x**2*ln(x)-180*x**3-108*x**2-25*x+5)/((4*x*ln(x)-2 0*x**2-12*x)*ln(-ln(x)+5*x+3)**2+((48*x**2-20*x)*ln(x)-240*x**3-44*x**2+60 *x)*ln(-ln(x)+5*x+3)+(144*x**3-120*x**2)*ln(x)-720*x**4+168*x**3+360*x**2) ,x)
Exception raised: PolynomialError >> 1/(-_t0*x + 5*x**2 + 3*x) contains an element of the set of generators.
Time = 0.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int \frac {5-25 x-108 x^2-180 x^3+36 x^2 \log (x)+\left (15-11 x-60 x^2+(-5+12 x) \log (x)\right ) \log (3+5 x-\log (x))+(-3-5 x+\log (x)) \log ^2(3+5 x-\log (x))}{360 x^2+168 x^3-720 x^4+\left (-120 x^2+144 x^3\right ) \log (x)+\left (60 x-44 x^2-240 x^3+\left (-20 x+48 x^2\right ) \log (x)\right ) \log (3+5 x-\log (x))+\left (-12 x-20 x^2+4 x \log (x)\right ) \log ^2(3+5 x-\log (x))} \, dx=-\frac {1}{4} \, \log \left (6 \, x + \log \left (5 \, x - \log \left (x\right ) + 3\right )\right ) + \frac {1}{4} \, \log \left (6 \, x + \log \left (5 \, x - \log \left (x\right ) + 3\right ) - 5\right ) + \frac {1}{4} \, \log \left (x\right ) \]
integrate(((log(x)-5*x-3)*log(-log(x)+5*x+3)^2+((12*x-5)*log(x)-60*x^2-11* x+15)*log(-log(x)+5*x+3)+36*x^2*log(x)-180*x^3-108*x^2-25*x+5)/((4*x*log(x )-20*x^2-12*x)*log(-log(x)+5*x+3)^2+((48*x^2-20*x)*log(x)-240*x^3-44*x^2+6 0*x)*log(-log(x)+5*x+3)+(144*x^3-120*x^2)*log(x)-720*x^4+168*x^3+360*x^2), x, algorithm=\
-1/4*log(6*x + log(5*x - log(x) + 3)) + 1/4*log(6*x + log(5*x - log(x) + 3 ) - 5) + 1/4*log(x)
Time = 0.35 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int \frac {5-25 x-108 x^2-180 x^3+36 x^2 \log (x)+\left (15-11 x-60 x^2+(-5+12 x) \log (x)\right ) \log (3+5 x-\log (x))+(-3-5 x+\log (x)) \log ^2(3+5 x-\log (x))}{360 x^2+168 x^3-720 x^4+\left (-120 x^2+144 x^3\right ) \log (x)+\left (60 x-44 x^2-240 x^3+\left (-20 x+48 x^2\right ) \log (x)\right ) \log (3+5 x-\log (x))+\left (-12 x-20 x^2+4 x \log (x)\right ) \log ^2(3+5 x-\log (x))} \, dx=-\frac {1}{4} \, \log \left (6 \, x + \log \left (5 \, x - \log \left (x\right ) + 3\right )\right ) + \frac {1}{4} \, \log \left (6 \, x + \log \left (5 \, x - \log \left (x\right ) + 3\right ) - 5\right ) + \frac {1}{4} \, \log \left (x\right ) \]
integrate(((log(x)-5*x-3)*log(-log(x)+5*x+3)^2+((12*x-5)*log(x)-60*x^2-11* x+15)*log(-log(x)+5*x+3)+36*x^2*log(x)-180*x^3-108*x^2-25*x+5)/((4*x*log(x )-20*x^2-12*x)*log(-log(x)+5*x+3)^2+((48*x^2-20*x)*log(x)-240*x^3-44*x^2+6 0*x)*log(-log(x)+5*x+3)+(144*x^3-120*x^2)*log(x)-720*x^4+168*x^3+360*x^2), x, algorithm=\
-1/4*log(6*x + log(5*x - log(x) + 3)) + 1/4*log(6*x + log(5*x - log(x) + 3 ) - 5) + 1/4*log(x)
Timed out. \[ \int \frac {5-25 x-108 x^2-180 x^3+36 x^2 \log (x)+\left (15-11 x-60 x^2+(-5+12 x) \log (x)\right ) \log (3+5 x-\log (x))+(-3-5 x+\log (x)) \log ^2(3+5 x-\log (x))}{360 x^2+168 x^3-720 x^4+\left (-120 x^2+144 x^3\right ) \log (x)+\left (60 x-44 x^2-240 x^3+\left (-20 x+48 x^2\right ) \log (x)\right ) \log (3+5 x-\log (x))+\left (-12 x-20 x^2+4 x \log (x)\right ) \log ^2(3+5 x-\log (x))} \, dx=\int \frac {25\,x-36\,x^2\,\ln \left (x\right )+{\ln \left (5\,x-\ln \left (x\right )+3\right )}^2\,\left (5\,x-\ln \left (x\right )+3\right )+\ln \left (5\,x-\ln \left (x\right )+3\right )\,\left (11\,x-\ln \left (x\right )\,\left (12\,x-5\right )+60\,x^2-15\right )+108\,x^2+180\,x^3-5}{\ln \left (x\right )\,\left (120\,x^2-144\,x^3\right )+\ln \left (5\,x-\ln \left (x\right )+3\right )\,\left (\ln \left (x\right )\,\left (20\,x-48\,x^2\right )-60\,x+44\,x^2+240\,x^3\right )+{\ln \left (5\,x-\ln \left (x\right )+3\right )}^2\,\left (12\,x-4\,x\,\ln \left (x\right )+20\,x^2\right )-360\,x^2-168\,x^3+720\,x^4} \,d x \]
int((25*x - 36*x^2*log(x) + log(5*x - log(x) + 3)^2*(5*x - log(x) + 3) + l og(5*x - log(x) + 3)*(11*x - log(x)*(12*x - 5) + 60*x^2 - 15) + 108*x^2 + 180*x^3 - 5)/(log(x)*(120*x^2 - 144*x^3) + log(5*x - log(x) + 3)*(log(x)*( 20*x - 48*x^2) - 60*x + 44*x^2 + 240*x^3) + log(5*x - log(x) + 3)^2*(12*x - 4*x*log(x) + 20*x^2) - 360*x^2 - 168*x^3 + 720*x^4),x)
int((25*x - 36*x^2*log(x) + log(5*x - log(x) + 3)^2*(5*x - log(x) + 3) + l og(5*x - log(x) + 3)*(11*x - log(x)*(12*x - 5) + 60*x^2 - 15) + 108*x^2 + 180*x^3 - 5)/(log(x)*(120*x^2 - 144*x^3) + log(5*x - log(x) + 3)*(log(x)*( 20*x - 48*x^2) - 60*x + 44*x^2 + 240*x^3) + log(5*x - log(x) + 3)^2*(12*x - 4*x*log(x) + 20*x^2) - 360*x^2 - 168*x^3 + 720*x^4), x)